We study the following chemotaxis-Navier–Stokes system with nonlinear production: \(n_t+u\cdot \nabla n=\Delta n-\nabla \cdot (n\chi (n)\nabla c)\) , \(c_t+u\cdot \nabla c=\Delta c-c+n^\beta \) , \(u_t+(u\cdot \nabla )u=\Delta u+\nabla P+n\nabla \Phi \) , and \(\nabla \cdot u=0\) in a bounded domain \(\Omega \subset \mathbb {R}^2\) , where \(\chi \in C^2([0,\infty ))\) , \(\beta >0\) , and \(\Phi \in W^{2,\infty }(\Omega )\) . In our previous work [Z. Angew. Math. Phys. 75 (2024) 74], it was shown that if there exists a certain \(k>0\) such that \(\begin{aligned} \left\{ \begin{aligned}&\chi (s)(s+1)^{-\frac{1}{2}}=O((s+1)^{-k})~\text{ as }~s\rightarrow \infty \quad & \text{ for }~0<\beta <\frac{1}{2}, \\&\chi (s)(s+1)^{\beta -1}=O((s+1)^{-k})~\text{ as }~s\rightarrow \infty & \text{ for }~\beta \ge \frac{1}{2}, \end{aligned} \right. \end{aligned}\) then for any suitably smooth initial datum, the corresponding initial-boundary value problem possesses a unique globally bounded classical solution. Especially, in the case that \(\beta \in (0,\frac{1}{2})\) , the restriction on the growth of the sensitivity \(\chi \) stems from the effect of fluid motion governed by the Navier–Stokes equations. In the present paper, we further indicate that when \(\beta \in (0,\frac{1}{2})\) , the solutions still remain globally bounded even if the cross-diffusion is intensified to satisfy \(\begin{aligned} \chi (s)(s+1)^{-\frac{1}{2}}=o(1) \quad \text{ as }~s\rightarrow \infty . \end{aligned}\) This improvement is obtained by more exhaustively controlling the destabilizing action of fluid-driven transport and so further reflects the underlying impact of fluid flow on global solvability.
